Metric:
We use the CNOT/single-qubit gate count, and the circuit depth in the post-compilation program to evaluate Paulihedral.
For the SC backend, the CNOT gate count is more important due to its higher error rate and latency. The depth is also important due to short qubit coherence times.
For the FT backend, T gate is usually more expensive but for the simulation kernels, the ratio between the H, Y, CNOT gate count and the T gate count grows linearly as the number of qubits increases.
Because a Pauli string of length \(n\)  will have \(O(n)\)  H, Y, and CNOT gates but the number of Rz gates (the only source of T gates) is always one.
It has also been shown that CNOT count is a significant cost in fault-tolerant algorithms and should not be neglected compared to T gates [1]}.
Hence, we estimate the performance with total gate count and circuit depth, following convention in previous work [2]}, [3]}, [4]}, [5]}.
